On the Mechanisms of Condensation and Evaporation in Disordered

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On the Mechanisms of Condensation and Evaporation in Disordered Porous Materials Fabien BONNET, Mathieu Melich, Laurent Puech, Jean-Christian Angles d'Auriac, and Pierre-Etienne Wolf Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b04275 • Publication Date (Web): 13 Mar 2019 Downloaded from http://pubs.acs.org on March 14, 2019

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On the Condensation and Evaporation Mechanisms in Disordered Porous Materials Fabien Bonnet,† Mathieu Melich,† Laurent Puech,†,‡ Jean-Christian Anglès d’Auriac,† and Pierre-Etienne Wolf∗,† † Université Grenoble Alpes, CNRS, Institut Néel, F-38042 Grenoble, France ‡Deceased 2008, Dec 2nd E-mail: [email protected] Abstract Sorption isotherms measurements is a standard method for characterizing porous materials. However, such isotherms are generally hysteretic, differing between condensation and evaporation. Quantitative measurement of pore diameter distributions then requires to properly identify the mechanisms at play, a topic which has been and remains the subject of intensive studies. In this paper, we compare high precision measurements of condensation and evaporation of helium in Vycor, a prototypical disordered porous glass, to a model incorporating mechanisms at the single pore level through a semi-macroscopic description, and collective effects through lattice simulations. Our experiment determines both the average of the fluid density through volumetric measurements, and its spatial fluctuations through light scattering. We show that the model consistently accounts for the temperature dependence of the isotherm shape and of the optical signal over a wide temperature range, as well as for the existence of thermally activated relaxation effects. This demonstrates that the evaporation mechanism evolves from pure invasion percolation from the sample’s surfaces at the

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lowest temperature to percolation from bulk cavitated sites at larger temperatures. The model also shows that the experimental lack of optical signal during condensation does not imply that condensation is unaffected by network effects. In fact, these effects are strong enough to make most pores to fill at their equilibrium pressure, a situation deeply contrasting the behavior for isolated pores. This implies that, for disordered porous materials, the classical Barrett-Joyner-Halenda approach, when applied to the condensation branch using an extended version of the Kelvin equation, should properly measure the true pore diameter distribution. Our experimental results support this conclusion.

INTRODUCTION The Barrett-Joyner-Halenda (BJH) method 1 is routinely used to determine the pore diameter distribution of mesoporous materials from the measurement of sorption isotherms. It relies on the fact that, for a cylindrical pore of radius R, and a wetting fluid, the vapor pressure corresponding to phase equilibrium is shifted below the saturated vapor pressure, the shift increasing when the radius decreases. For wide enough pores (typically larger than 100 nm), this shift can be understood in terms of capillarity through Kelvin’s law. For narrower pores, the situation is more complex: the interaction with the substrate should be taken into account, as well as, below a few nanometers, the atomistic structure of the fluid and the substrate. Beyond these effects, the main difficulty to apply the BJH scheme is the existence of an hysteresis between condensation and evaporation below a critical temperature Tch , smaller than the bulk critical temperature Tc . Below Tch , a central question is then to identify which of the condensation and evaporation branches, if any, corresponds to a phase equilibrium. Thanks to the combination of experiments and analytical or numerical theoretical approaches, the situation is today well understood for single cylindrical or slit pores, or regular systems of pores (for reviews, see Refs. 2,3). For cylindrical pores, studies using local 4 or 2


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non local 5 density functional theories have shown that, above a diameter of about several nanometers, a macroscopic capillary description of the condensation and evaporation phenomena is reasonably accurate. For a perfectly wetting substrate and a cylindrical pore, condensation first proceeds by the adsorption of a film on the inner surface of the pore, due to the substrate attraction. This film becomes metastable with respect to the fully filled state above the equilibrium pressure corresponding to its coexistence with a liquid region, and fully unstable above a spinodal-like pressure. 6 In the absence of thermal activation, condensation in a single pore open at both ends then proceeds through the spinodal instability, while evaporation takes place at equilibrium by recession of the meniscus. In this situation, the equilibrium branch to be considered in the BJH theory is expected to be the evaporation one. However, coupling between different pores modify this simple picture. For a wide pore (cavity) surrounded by narrower pores (constrictions), condensation in the constrictions can trigger condensation of the cavity below its spinodal pressure (advanced condensation). Inversely, the energetical cost for creating a liquid-vapor interface blocks the evaporation inside the cavity at its equilibrium pressure. Evaporation is delayed until thermal activation allows the creation of such an interface (cavitation) or the constrictions themselves empty, creating a meniscus at the cavity entrance which can then recess. Both possibilities have been demonstrated in patterned silicon wafers, 7,8 and in ordered porous materials consisting of regular networks of cavities connected by narrow constrictions. 9–11 The case of disordered porous materials is more complex. If cavitation is negligible, evaporation should proceed by a collective invasion percolation mechanism from the surfaces. 12–14 Such a mechanism has been experimentally demonstrated by room temperature light scattering experiments on hexane in Vycor, a prototypical disordered glass consisting of interconnected cylindrical pores. 15 In the opposite case, it has been predicted that evaporation could also proceed by invasion from the cavitated pores. 16 In both cases, the pore diameter distribution cannot be simply retrieved from the evaporation branch of the 3


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isotherm. Experimentally, based on an analysis of the evolution of the isotherms shape with temperature, 17 a temperature driven crossover between pure percolation and cavitation has been reported for evaporation of nitrogen in Vycor. However, later measurements using helium as a fluid and light scattering as a probe have suggested that the temperature dependence of the scattered intensity was a better marker of a crossover than that of the isotherms shape 18 . On the theoretical side, local mean field density functional theory (LMF-DFT) has been successfully applied to describe condensation and evaporation in disordered porous materials 19 such as silica aerogels and Vycor. In the first case, the predicted behaviour 20 accounts well for later measurements. 21 Calculations on Vycor 22 based on a realistic description of the actual porous structure of Vycor remarkably reproduce not only the experimental isotherms shape and its temperature dependence, but also the observed shape of scanning curves. Furthermore, Monte Carlo simulations based on the same description do evidence the occurrence of cavitation for specific simulation parameters. 23 Despite this impressive success, two reasons justify to explore an alternative theoretical description. First, the LMF-DFT approach assumes a known microstructure. Inferring pores sizes for an arbitrary porous material would thus have a very high computational cost. Second, at least in its present state, LMF-DFT does not provide a general physical picture of the interplay between basic mechanisms. In this paper, we develop such a picture based on a multiscale approach, benchmark it against experimental measurements, and discuss how its results can be used to obtain pore diameter distributions from sorption isotherms measurements.

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METHODS Experimental We summarize here our experimental methods. Details can be found in the Supporting Information and in Refs. 18,24. Our sample is a 14 mm diameter, 4 mm thick, disk of Vycor 7930, contained in an annular copper cell closed by two sapphire windows. The cell was regulated between 3 K and Tc ≈5.2 K in an optical cryostat. 18 Condensation and evaporation of helium were controlled by sweeping the temperature of a gas reservoir connected to the cell. 25 The cell pressure, the confined fluid density, and the corresponding liquid fraction were obtained as described in the Supporting Information. The high accuracy of these measurements was central to study in detail the shape and closure of the hysteresis loop. Isotherms were measured for about 20 temperatures ranging from 3 K up to 5.22 K, slightly above Tc =5.195 K. We typically scan the condensation or evaporation branch of the hysteresis loop in 10 to 20 hours below 4 K, and several hours near the closure temperature of the hysteresis loop, Tch ≈ 4.65 K. We checked such flowrates to be small enough to avoid thermal gradients within the sample due to the heat released (resp. absorbed) by the condensation (resp. evaporation) process. Relaxation effects due to thermal activation were studied by stopping the flowrate at various filling fractions and measuring the pressure relaxation at a fixed reservoir temperature over several hours. Sorption isotherms only give access to the fluid density averaged over the Vycor sample. Spatially resolved information was obtained by simultaneous light scattering measurements. The sample was illuminated with a thin laser sheet at a wavelength of 632 nm under 45◦ incidence with respect to the common disk and windows axis, with a polarization perpendicular to the scattering plane. The illuminated slice was imaged using two CCD cameras at opposite observation angles (45◦ or 135◦ from the incident direction). In our regime of single scattering, the signal of each CCD pixel measures the light scattered in the direction of observation by the corresponding imaged region, and probes the static spatial fluctuation 5


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of the fluid density at the considered point, on a spatial scale set by the observation angle.

Modelization Description of the model

In order to modelize the effect of connections between pores,

we combine a realistic calculation of the condensation and evaporation processes within a single cylindrical pore 26 with a lattice description of the pores network. This approach compromises between those respectively used by Guyer and McCall 27 to discuss evaporation by percolation in 3D disordered porous networks, and by Puibasset 28 to model condensation and evaporation in a single corrugated cylindrical pore. Our treatment of the single pore is more realistic than in Guyer and McCall’s approach, as it includes in particular the key effect of thermal activation. On the other hand, being based on the assumption that, locally, the fluid obeys the bulk equation of state, it is less exact than the Grand Canonical Monte Carlo approach used by Puibasset. However, the gain in simplicity allows to modelize a complex 3D network. Our approach, while similar in spirit, differs from that used in Ref. 16 by two important aspects. At the single pore level, it takes into account the size dependence of the homogeneous cavitation, 18 a crucial fact not recognized by the authors of Ref. 16. At the network level, we consider a cubic rather than a Bethe lattice. While this does not allow for analytical calculations, it provides a more realistic description of an actual porous material. Our single pore model predicts three temperature dependent characteristic pressures for a cylindrical pore of radius R. These pressures are respectively: (i) Peq (R), the pressure corresponding to the liquid-film coexistence within the pore, (ii) Pact (R), the pressure at which the pore fills by thermally activated nucleation of a liquid bridge, and (iii) Pcav (R), the cavitation pressure below which a pore empties by thermally activated nucleation of a vapor bubble. These pressures must be understood as the vapor pressure of the reservoir (far away from the pore walls). They are obtained from the corresponding fugacities through the equation of state of helium. The latter are predicted by the model from the fluid properties at the temperature T , the pore radius, the van der Waals attraction between the fluid and 6


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the substrate, and the barrier height which can be overcome on the experimental time scale. For simplicity, we will speak below of pressure P rather than fugacity, but it should be remembered that, for all theoretical curves, P is actually a fugacity, P =1 corresponding to the saturated vapor pressure Psat . Accordingly, comparisons of model and experiment will be made by converting the experimental pressures to fugacities. (a) 0.75 0.5

T=4

0.25

(b)

Pact Peq Pcav

0 0.75 P

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0.5

T=3.5

0.25 0 0.75 0.5

T=3

0.25

R(nm) 1 2

4

6

8

10

Figure 1: (a) The three single pore characteristic reduced pressures (actually fugacities) Pcav , Peq , Pact . P = 1 corresponds to the saturated vapor pressure Psat . Their dependence on the pore radius, shown for three temperatures, is computed from Ref. 26 and used as an input for the coupled model. The bullet corresponds to the temperature dependent critical radius Rc (T ) and the triple point pressure P3 . (b) Modelization of the pore network by a cubic lattice of sites occupied by hexapod building blocks. The picture shows two adjacent building blocks meeting at the middle of the bond. Figure 1a shows the radius dependence of these pressures for three temperatures, computed for a van der Waals interaction of 1200 KÅ3 when a barrier 38 kB T high can be overcome. Below a critical radius Rc (T ) and a pressure P3 , the three pressures coincide, corresponding to a a reversible condensation-evaporation process. Above Rc (T ), which increases with temperature, the three pressures differ. For large enough pores, Pcav (R) approaches the bulk cavitation pressure Pcav (∞), while Pact (R) and Peq (R) tend to unity. Note that Pact (R) and Pcav (R) behave quite differently when thermal activation is inhibited by decreasing the temperature. While cavitation is suppressed, condensation below the saturated vapor pressure remains possible through the spinodal instability of the adsorbed film. 6,26 This difference 7


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will turn out to be responsible for the different optical behavior between evaporation and condensation. For the same pore within a network, the actual evaporation and condensation mechanisms depend on the pores surrounding. The coupling mechanisms described in the introduction can be translated into four evolution rules. The two first ones apply whatever the state of the pore’s neighbor, while coupling comes into play for the two last ones. • (i) For P ≤ Pcav (R), the pore is empty (but for the adsorbed film) • (ii) For P ≥ Pact (R), the pore is filled • (iii) For Peq (R) < P < Pact (R), a pore fills and only if it has at least a filled neighbor. • (iv) For Pcav (R) < P < Peq (R), it empties if and only if it has at least an empty neighbor. The two first rules give rise to hysteresis if T is smaller than the hysteresis closure temperature Tch , such that the critical radius Rc (Tch ) is the radius of the largest pore. Above Tch , Pcav = Peq = Pact , and the sample’s configuration is simply determined by rules (i) and (ii). Below Tch , the four rules allow to determine the system state for any given pressure history. Our model is thus described, as many out-of-equilibrium models, by a cellular automaton rather than by a hamiltonian. In so far as the barrier height which can be overcome is fixed, Pcav (R) and Pact (R) do not depend on time so that our model has no relaxation dynamics. These rules can be applied to any geometry of the pores network. To be specific, we consider a cubic lattice of lattice constant a and linear size L = N a with N =512. a is a free parameter, which is expected to be of order the experimental correlation length of the Vycor pore network. Each site of the lattice is occupied by the hexapod unit shown in Figure 1b. The pores of the hexapod have a radius randomly drawn at each site according to a gaussian distribution. The pores radii are thus independent identically distributed (iid) 8


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variables. The length of each pore of the hexapod being equal to a/2, the volume of an hexapod unit scales as R2 . For the sake of simplicity, we will ignore the adsorbed film and the liquid compressibility, so that each hexapod can only be in two states, empty (0) or filled (1). Contact with the vapor phase at the sample’s surfaces is simulated by adding an extra layer of sites around the sample and fixing their state to 0. Alternatively, we can suppress the influence of the surface by using periodic boundary conditions. We can also probe the respective roles of cavitation and pore coupling by comparing the full solution to those respectively obtained forcing Pcav to zero , or disregarding rules (iii) and (iv). For the chosen geometry, neighboring pores are directly coupled (Figure 1b). In contrast, former studies 13 have considered large cavities (sites) separated by narrow windows (bonds). Evaporation in the absence of thermal activation is then a bond percolation problem, instead of a site percolation problem in our model. Beyond this difference, our model is oversimplified with respect to real Vycor. First, pores of different radii only branch axially, and not laterally. Second, for our cubic lattice, a pore has six neighbors, a larger number than suggested from TEM pictures. 29 Third, neighboring pores have uncorrelated diameters. These simplifications will affect the percolation threshold, so that we cannot hope for a detailed quantitative agreement betwen the model and the experiment. However, as we will show, our description allows to capture the qualitative features of the experimental results, in particular the temperature evolution of the optical signal. Obtaining the fluid configuration

In order to predict the condensed fraction and the

optical signal for given pressure history and temperature, we first have to determine the configuration for the target pressure. On that aim, we can use two equivalent numerical methods. The first one, used in the present work, is described in the Supporting Information(§S3.1). Here, we describe an alternate method, based on the concept of Bernoulli clusters, as this concept is useful to discuss our numerical results. We consider only the case of a monotonous pressure history, but, as described in the Supporting Information, the

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method can be generalized to an arbitrary one. The method starts by marking, at the target pressure of interest P , the sites potentially filled for condensation (respectively empty for evaporation), that is the sites of radius R such that Peq (R) < P (respectively Peq (R) > P ). We then group these sites into maximally connected clusters (the so-called Bernoulli clusters) by using a fast algorithm. 30 According to the coupling rules (iii) or (iv) above, all sites within in given cluster are in the same state. Among these potential clusters, the effectively filled clusters for condensation are those containing at least a site with Pact (R) < P . According to rule (ii), this site (denominated germ below) will be filled (state 1), thus triggering the filling of the full cluster. For evaporation, the effectively empty clusters are those containing a vapor site (state 0). Such sites are either the boundary sites, or the cavitated sites (Pcav (R) > P ). Correspondingly, we have two kinds of germs and two selections rules instead of one for condensation. Once identified the relevant germs at pressure P , we find the corresponding selected clusters, hence the configuration. In the remaining of this paper, clusters of sites potentially filled (for condensation) or empty (for evaporation) will be referred to as potential clusters, while the clusters effectively filled (respectively empty) will be refereed to as selected clusters. For both condensation and evaporation, some cluster can percolate throughout the sample when the fraction of sites potentially filled or empty reaches 31%, the percolation threshold for site percolation on a 3D cubic lattice. Introducing Rperc , the radius such that 31% of the pores have a larger radius, we define the percolation pressure for evaporation Pperc = Peq (Rperc ). Computation of isotherms

From the configuration, we obtain the liquid total amount

as a function of the external pressure, i.e. the isotherm. In our lattice model, the volume of a pore is proportional to R2 , but other distributions could be chosen to evaluate the impact of the actual geometry for a fixed connectivity. As explained above, we neglect the film contribution and the effect of the liquid compressibility. These approximations little affect the isotherms shape, so that qualitative, if not quantitative, comparison to the

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experimentally measured isotherms will be possible. Computation of structure factor

In order to compare our model to our light scattering

measurements, we also compute the structure factor of the local liquid density. Within our lattice model, this quantity, to which the scattered intensity is proportional, is obtained from the fluid configuration as: S(q) = |

X

F (n) exp(ıqna)|2

(1)

nx,y,z

Here, n is the site of integer coordinates (nx , ny , nz ) (nx , ny , nz run between 0 and N − 1), q =(qx , qy , qz ) is the transfer wavevector, defined by the directions of incidence and observation and the light wavelength, a is the lattice constant, and F (n) is the scattering amplitude at site n. 31 We approximate F (n) as : F (n) = a[(a2 − CRn2 )αSi + CRn2 δn αHe ]

(2)

where Rn is the site radius, CaRn2 the associated void volume, δn = 0 (resp. δn = 1) if the site is empty (resp. filled), and αSi and αHe are the silica and helium polarisabilities, respectively related to the corresponding dielectric constants through the Clausius-Mosotti equation, α=

−1 . +2

Since =1.057 for helium and 2.4 for silica, we use αSi ≈ 1 and αHe ≈ 0.05. For

non zero q, S(q) reduces to S(q) ∝ |

X n

Rn2 (αSi − δn αHe ) exp(ıqna)|2

(3)

Because Rn is randomly distributed, this expression, in contrast to that corresponding to the simpler choice Fn = δn , accounts for the finite scattering by the bare silica network (δn = 0). It also accounts for the experimentally observed reduction, due to index matching, of the scattered intensity for the filled sample (δn = 1). For computation, we restrict q to the reciprocal lattice, thus allowing the use of discrete 11


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Fourier transform. We thus write q = 2π NQa , where Qx , Qy , Qz are integers between 0 and N -1. The modulus Q of Q then corresponds to the scattering angle θ through Q = 2N a sin(θ/2)/λ. S(Q), now independent on a, is evaluated using the Fast Fourier Transform algorithm provided at www.fftw.org. The result strongly fluctuates as a function of Q, corresponding to the so-called speckle phenomenon. In an actual structure factor measurement, these fluctuations are washed out by using a detector covering many speckle spots around a given scattering direction. Therefore, we average S(Q) over the direction of Q to keep only the modulus dependence. To this aim, we sum the computed structure factor over all Q vectors located in a shell of √ √ ˜ + 1/2) 3, where Q ˜ is integer, yielding an isotropized S(Q). 32 width 3 centered on Q = (Q The final optical signal we will compare to the experiment is the real piecewise constant function 2 X 1 S(Q) = S(Qi ) N (Q) where the sum runs over the shell, and N (Q) is the number of points in the shell. In the case where the states of the sites are not correlated, for example above Tch where Pcav = Pact = Peq for all sites, the δn ’s are iid random variables, so that hS(q 6= 0)i = hF (n)2 i − hF (n)i2 . Because αSi αHe , this expression is dominated by the silica signal, and hardly depends on the liquid fraction. Obtaining a signal larger than for bare silica is only possible if the site occupancy is spatially correlated. As we will show, this requires not only pores coupling, but also a selection process of the resulting clusters.

RESULTS AND DISCUSSION Experimental results

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140

density (g/L)

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120 3.4 K 3.75 K 4.34 K 4.6 K 4.9 K 5.1 K

100

0.7

0.8

0.9 P/Psat

Figure 2: Selected condensation-evaporation isotherms of helium in Vycor from 3.4 K up to above Tc , showing the average density of the confined fluid as a function of P/P0 , the pressure relative to the saturated pressure. The hysteresis loop changes in shape and extent as the temperature is increased. It closes between 4.6 and 4.7 K, well below the bulk critical temperature. Isotherms measurements Figure 2 shows the temperature evolution of isotherms from around 3.4 K to above Tc . Below approximately 4.4 K, the hysteresis loop is triangular, the evaporation branch presenting a sharp kink followed by a steep section as the pressure is decreased. Such a H2-type behavior has been previously reported in Vycor for a large variety of fluids (for example argon, 33,34 nitrogen, 17,34,35 hexane, 15 water 36 ), and often discussed as an evidence for the invasion percolation evaporation mechanism. 12–14 Above 4.4 K, the loop becomes less triangular, and, at a flowrate of 0.3 STPcc/min, closes around 4.7 K (see Figure S2 for a zoom). This closure temperature Tch is about 10% lower than the bulk critical temperature Tc . It is larger than previously observed by Brewer and Champeney 37 for helium in Vycor, who found no hysteresis at 4.2 K. This difference could be due to a less good resolution in their case. 38 Loop closure below Tc has been previously reported in Vycor for other fluids (nitrogen, 17 carbon dioxyde 39 ) or other porous glasses. 40,41 However, to our knowledge, our experiment is the first one to follow the loop closure with such a high accuracy. In particular, we clearly show that the evaporation isotherm becomes less steep as temperature increases up to Tch , 13


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in contrast to the previous conclusion of Morishige for nitrogen in Vycor. 17 Above Tch , the isotherm is fully reversible, with no sharp kink, but an elbow which becomes fainter as the temperature increases up to Tc . Based on theoretical calculations for a distribution of independent pores, 26 we interpret this elbow as the pressure where all pores are fully filled with liquid. Two mechanisms can account for the closure of the hysteresis loop at Tch < Tc . The first one is a shift of the critical temperature due to confinement 42 which would suppress the surface energy, hence the energy barrier. The second one is thermal activation, which would allow to overcome the energy barrier at equilibrium, as first proposed by Machin. 41 Similarly to Burgess et al, 39 we can extract from our isotherms measurements a ”coexistence curve” representing, as a function of temperature, the fluid density at the lower and upper ends of the hysteresis loop. As discussed in the Supplemental Information, this analysis shows that the hysteresis loop is non universal, unlike expected for a confinement induced shift of Tc . 42 In contrast, our measurements are consistent with a thermally activated closure mechanism. First, according to our single pore modelization (Figure 1a), a closure temperature at Tch ≈ 4.7 K corresponds to a maximal pore radius of order 7 nm. As we will find below, this value agrees with the measured pore diameter distribution (Figure 10). Second, the thermal activation scenario is directly supported by the relaxation measurements we now describe. Relaxation effects

Relaxation effects along the hysteresis loop were studied as described

in the methods section. When the reservoir temperature sweep is stopped at some fixed set point within the hysteresis loop, we observe a following slow relaxation of the pressure. On the evaporation branch, the pressure increases, corresponding to the evaporation of the cell towards the reservoir, and reversely on the condensation branch. Figure 3 shows such measurements for a temperature of 4.34 K. The relaxation is obvious on the evaporation branch, while it hardly visible on the condensation branch 1 . The inset of Figure 3 shows 1

As discussed in section 4.3, the small relaxation on the condensation branch is consistent with the idea that, due to pore coupling, condensation takes place close to equilibrium.

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136 T=4.34 K evaporation condensation 10 129

∆ P (mb)

ρ (g/L)

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10-3 122 0.9

0.92

0

time (hours)

0.94 P/Psat

30

0.96

Figure 3: Relaxations of pressure and fluid density within the hysteresis loop at 4.34 K, recorded after stopping the condensation or the evaporation. The inset shows the time dependence of the pressure relaxation. that the relaxation timescale is longer than several hours for both branches. This long timescale, and the fact that relaxation is only observed within the hysteresis loop, show that it cannot be due to the transport of mass or heat within the sample. As discussed in the Supporting Information, relaxations are observed for temperatures ranging from 3.16 K up to Tch , and are consistent with thermally activated condensation and evaporation for cylindrical pores distributed in diameter. In the temperature region where the loop closes, relaxation makes Tch dependent on the waiting time (see Figure S3 of the Supporting Information). This strengthens the above interpretation of the hysteresis closure in terms of a thermally activated mechanism. We note that long time scale relaxations for condensation and/or evaporation in Vycor have also been reported using respectively ultrasonic or NMR measurements. 15,43 The relaxing variables being different, we cannot determine whether these relaxations are related or not. NMR relaxations were interpreted as the signature of a thermally activated reorganization of the adsorbate. In our case, the temperature and pressure dependence of the relaxation times is consistent with the idea that this reorganization takes place on a local scale, at the level of the single pore.

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Optical signal

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The evolution of the isotherm shape with temperature can be qualitatively

explained by assuming that condensation and evaporation take place independently in each pore. 18 This is surprising as Vycor’s pores are highly connected, and coupling effects are to be expected. As we now show, light scattering measurements do reveal such coupling effects during evaporation.

1

2 1

Φ 3

0.9

4

0.8

0.7 5 0.6 0.68

P/Psat

0.76

Figure 4: Successive (false color) images of a Vycor slice during evaporation at 3 K. The slice is imaged at 45◦ from the illuminating light sheet. The positions of the different images on the evaporation isotherm are indicated on the graph. Averaging the brightness over the white rectangle gives the optical signal shown in Figure 5a. Figure 4 shows images of the Vycor sample during evaporation at 3 K. In the initial state (image 1), the sample is filled and the light is only scattered by the small scale silica heterogeneities. Right at the kink in the isotherm, the sample gets slightly brighter. As evaporation proceeds further, the brightness increases nearly uniformly, reaching a maximum at an average liquid fraction Φ around 0.9. At the lower end of the hysteresis loop (image 5), the image is slightly brighter than for the filled sample. The difference reflects the index matching by helium. At its maximum, the image brightness is about twice that for the filled sample. Because the dielectric constant of helium is much smaller than that of silica, this implies that the vapor distribution is correlated over distances much larger than the silica correlation length. Such long range correlations are in agreement with the fact that, at maximum, the scatter16


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2

o

2

1.6 I/ISi0

(a)

T= 3 K (E-45o) T= 3 K (E-135o) T= 3.16 K (E-45o) T= 3.16 K (E-135 ) o T=3.4 K (E-45 ) T=3.75 K (E-45oo) T=3 K (C-45 ) T=3.75 K (C-45o)

1.8

1.4 1.2 1 0.8

0.6

0.7

0.8 Φ

0.9

1

2 T= 3.2 K (E-5) T= 3.2 K (E-12) T= 3.4 K (E-5) T= 3.4 K (E-12) T= 3.5 K (E-5) T= 3.5 K (E-12) T= 3.7 K (E-5) T= 3.7 K (E-12) T= 4.3 K (E-5) T= 4.3 K (E-12) any T and Q (C)

1.8

2

1.6 I/ISi0

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1.4 1.2

(b)

1 0.8 0

0.2

0.4

0.6

0.8

1

Φ

Figure 5: Comparison of (a): the measured and (b): the predicted optical signals for condensation (C) and evaporation (E) between 3 K and 4.34 K. The signals are plotted versus ˜ the liquid fraction Φ for two scattering angles (θ = 45o and θ = 135o , corresponding to Q=5 o and 12 for the model). θ = 135 data are shown only when they differ from the θ = 45o data. ing is slightly forward-peaked, being larger for 45◦ than for 135◦ . When the temperature increases, the brightness enhancement gradually decreases, and scattering becomes isotropic, consistent with a decrease of the correlation range. This is quantified in Figure 5a, which shows how the scattered intensity, averaged over the rectangle shown in Figure 4, evolves with the liquid fraction from 3 K to 3.75 K. 44 In contrast, in the same temperature range, the scattered intensity during condensation continuously decreases from the empty Vycor value to the index matched value in the liquid state. This difference between condensation

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Page 18 of 34

and evaporation disappears for larger temperatures, i.e. at 4.34 K and beyond. In this temperature range, the dependence on fraction of the scattered intensity is the same for condensation and evaporation, even though the isotherm is hysteretic below 4.6 K. This temperature evolution of the scattered intensity is a result of cavitation. Because, at all temperatures, the pressure Peq (R) monotonously increases with the radius R, the sequence of potential clusters during evaporation does not depend on temperature. If evaporation would only proceed by vapor percolation from the sample’s surfaces, 13 the selection rule will also be temperature independent, hence the successive configurations and the optical signal would be identical for any temperature, in contrast to our observations. Light scattering measurements were also performed during the above described relaxation experiments. At the lowest studied temperature (3.16 K), we found that, close to the kink of the evaporation isotherm, the optical signal grows over the same long time period during which the pressure relaxes (see Supporting Information). This shows that this signal is cavitation driven. However, evaporation by cavitation alone cannot produce an optical signal, in contrast to what was apparently assumed in the LMF-DFT study of evaporation in Vycor. 23 Indeed, for pure cavitation, the vapor sites would not be spatially correlated. As explained in the modelization methods section, this would result in a negligible dependence of the optical signal on the liquid fraction. Hence, our experiment demonstrates that both pores coupling and cavitation should be taken into account to properly describe evaporation in Vycor at all temperatures. In the following, we show that our model incorporating both effects indeed accounts for all our experimental observations.

Analysis Comparison of experimental results to simulations

Figure 6 compares the exper-

imentally measured isotherms to those computed within the model. For the experimental curves, the pressure scale of Figure 2 is converted to fugacity using the bulk equation of state of helium. The theoretical curves have been computed using a gaussian distribution 18


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1

(a)

Φ

0.9

0.8

0.7 3.16 K 3.75 K 4.34 K

0.6 0.6

1

0.7

0.8 fugacity

0.9

1

(b)

0.8 0.6 Φ

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0.4 0.2

3.2 K 3.8 K 4.3 K

0 0.6

0.7

0.8

0.9

1

P

Figure 6: Comparison of (a); selected experimental isotherms to (b); the model. For both (a) and (b), the liquid fraction Φ is plotted as a function of fugacity. For the theoretical max . isotherms, horizontal arrows indicate Pperc and vertical arrows Pcav truncated to positive values, with respective mean and width 3 nm and 1.4 nm chosen to approximately reproduce the characteristic pressures for condensation and evaporation, and the hysteresis closure temperature Tch ≈ 4.6 K. Figure 6 shows that our model properly reproduces the triangular (H2) shape of the hysteresis loop for the three temperatures shown. 45 The optical signal predicted by the model, for the same distribution as above, is plotted ˜ in Figure 5 for Q=5 and 12. These values correspond to the experimental angles of 45◦ and 135◦ when choosing a lattice constant a=15 nm, reasonably close to 30 nm, the measured correlation length of the Vycor structure 29 . Comparison of Figures 5a and 5b shows that the model remarkably accounts for the very different behavior of the optical signal during 19


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condensation and evaporation. Starting with condensation, it predicts the signal to be very weak, temperature and angle independent, and decreasing with fraction from its empty value down its lower filled value. This is in agreement with the experiment. Next, for evaporation, the model reproduces the four main features observed in the experiment : (i) for the largest temperatures (4.34 K and above), the identity of the signal to that observed for condensation ; (ii) for smaller temperatures, the gradual emergence of a peak of intensity for fractions around 0.8-0.9. (iii) the observation of an anisotropy at the lowest temperatures. (iv) the maximal enhancement of the signal, about twice the empty Vycor contribution at the lowest temperature (this specific value being fixed by the ratio αHe /αSi ). Beyond the difference in the fraction scale (due to the film contribution), the quantitative differences probably reflect the approximations made by the model. We conclude that our approach captures the important ingredients of the phenomena at play. We now discuss the underlying physics. Analysis of the model’s predictions

From the model, we can construct the P -T dia-

gram shown in Figure 8, which predicts the nature of the evaporation mechanism, depending on temperature and pressure. This diagram (which is detailed in the Supplemental Information) allows to define a crossover temperature Tco such that, below Tco , evaporation is essentially governed by percolation from the surfaces, and above, by percolation from the max , the cavitation pressure cavitated pores. Tco is defined by the intersection of two lines, Pcav max of the largest pores (for our distribution Pcav ≈ Pcav (∞)), and Pperc , the percolation pres-

sure where (potential) clusters contact to form a larger cluster percolating throughout the sample. For the parameters used above, Tco = 3.35 K. Figures 7a and b show hysteresis loops computed for two temperatures respectively below and above Tco , 3 K and 3.7 K. max Below Tco , evaporation above Pcav is fully controlled by the surfaces. Only the potential

clusters contacting these surfaces empty as the pressure is decreased. The liquid fraction

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1

(a)

Pmax cav

T= 3 K

Pperc

Liquid fraction Φ

0.8 0.6 0.4 with coupling no coupling flip at Peq no cavitation periodic

0.2 P3

0 0.5

0.6

0.7

0.8

0.9

P

1

(b)

T= 3.7 K

Pmax Pperc cav

0.8 Liquid fraction Φ

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0.6 0.4 P3 with coupling no coupling flip at Peq no cavitation periodic

0.2 0 0.6

0.7

0.8

0.9

1

P

Figure 7: Combined effect of pore coupling and thermal activation on the isotherm shape, computed from our model at two symmetric temperatures with respect to the percolationcavitation crossover at 3.35 K: (a) at 3 K, the evaporation is controlled by invasion from the surfaces down to Pmax cav . (b) at 3.7 K, the evaporation is controlled by invasion from the cavitated sites. In both cases, the condensation isotherm coincides with the equilibrium prediction. max starts decreasing approximately at Peq , as the clusters contacting the sample’s surfaces

grow in size, due to the increase of the correlation length for percolation, ξ. ξ being finite, their localized contribution however vanishes in the limit of an infinitely large sample. This corresponds to the dark pink region in Figure 8. In contrast, when the pressure reaches Pperc , ξ diverges, and a cluster percolates through the sample. For smaller P ’s, the number of sites in the percolating cluster rapidly grows and the fraction sharply drops (pink region in max Figure 8). Decreasing the pressure further, P reaches Pcav , and cavitation comes into play,

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4.5

Pmax cav Pperc

4

P3

~ PmaxO (Q=5) T

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3.5

reversible

TCO 3

n tio

ion

a vit

ca 2.5 0.4

0.5

surfaces

t ola

rc

pe

0.6

0.7 P

0.8

0.9

1

Figure 8: P -T diagram showing the different mechanisms of evaporation above and below Tco for a 5123 lattice (see text). The line PmaxO gives the pressure of maximal optical signal ˜ for Q=5. decreasing the liquid fraction with respect to the case of pure invasion from the surfaces (inset max of Figure 7a and white region in Figure 8). For P < Pcav , the inset of Figure 7a shows that

using periodic boundary conditions, hence suppressing invasion from surfaces, negligibly affects the liquid fraction. This is due to the fact that the largest cluster contacting the surfaces contains a macroscopic fraction of sites, hence one of the very first cavitated sites. Finally, below the pressure P3 of the triple point in Figure 1, all remaining pores empty reversibly at their equilibrium pressure (blue region in Figure 8). max Above Tco , Pperc < Pcav . Apart close to the sample’s surfaces, evaporation is driven

by cavitation (white region in Figure 8). In contrast to the situation below Tco , isotherms are then identical for periodic and non periodic conditions (Figure 7b). Coupling effects are nevertheless important, as cavitation of a given site selects the whole cluster to which it belongs. This explains why the evaporation branch of Figure 7b is steeper than for the max uncoupled case. The closer Pcav is to Pperc , the larger is the typical extent of the potential

clusters, and the larger the decrease of the liquid fraction with respect to the uncoupled case. Comparison of Figures 7a and b shows that the change of evaporation mechanism at Tco has no obvious signature in terms of the isotherm shape, which is of the H2-type both

22


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2

2

no cavitation T= 3.0 K Per-T= 3.2 K T= 3.3 K Per-T= 3.3 K T= 3.4 K Per-T= 3.4 K T= 3.5 K Per-T= 3.5 K T= 3.7 K Per-T= 3.7 K T= 3.8 K Per-T= 3.8 K condensation

1.5

I/ISi0

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~ Q=5

1

0.2

0.4

0.6 Φ

0.8

1

˜ = 5). Figure 9: Dependence of the theoretical optical signal on the evaporation process (for Q The temperature independent signal for condensation coincides with that computed without max coupling effects. Below the crossover temperature Tco =3.35 K, and above the pressure Pcav (indicated by arrows), the signal for evaporation is temperature independent and corresponds to percolation invasion from the sample boundaries. Above Tco , the temperature dependent signal corresponds to invasion from the cavitated sites. For each temperature, the liquid fraction at which percolation takes place is indicated by a dashed line. above and below Tco . In contrast, Figure 9 shows that that this change has a distinct optical max (T ), when signature. Below Tco , the optical signal only depends on temperature below Pcav

cavitation sets in. Above this pressure, a temperature independent peak occurs below the percolation pressure. The peak is angle dependent: it shifts towards smaller fractions and ˜ is increased (compare the curves at Q=5 ˜ pressures, and decreases in amplitude as Q and 12 at 3.2 K in Figure 5b). In contrast, above Tco , the peak occurs above the percolation pressure. Its position and amplitude are sensitive to the temperature. The peak amplitude ˜ is weaker than is reduced with respect to the low temperature case, and its dependence on Q below Tco (compare the curves at 3.2 K and 3.4 K in Figure 5b). As detailed in the Supplemental Information, these different features can be traced to max the light scattering properties of invasion percolation. Below Tco and above Pcav (T ), the

peak is controlled by the percolation cluster. Above Tco , it is due to the clusters selected from the cavitated germs. Starting from large liquid fractions, the optical signal grows due to the increase of the number and extent of these clusters, approximately up to the point 23


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where the average distance between germs equals the typical cluster extent. Beyond this point, the vapor density becomes more homogeneous and the signal decreases. Comparing Figure 5a to Figure 9, we see that the predicted saturation of the optical signal below 3.35 K is not experimentally observed. At 3.16 K, the signal is significantly smaller than at 3 K, and the anisotropy smaller. This points to the fact that, for our experimental system, Tco lies below 3.16 K (and possibly below 3 K). This is consistent with the observation of relaxation effects at the isotherm kink at the latter temperature (see Figure S8 in the Supporting Information). This discrepancy can be explained if the model of a cubic lattice overestimates the coordinance of the actual network. 3D networks with lower coordinance (3 or 4) have larger percolation thresholds, with critical fractions ranging between 40% and 60% instead of 31%. Such topologies would decrease the crossover temperature Tco below 3.35 K, and would better account for our experimental results. Let us now turn to condensation, for which coupling has a dramatic effect. Figures 7a and b indeed show that, at all temperatures, the condensation isotherm markedly differs from that in the uncoupled case, and nearly coincides with that computed assuming condensation to take place at equilibrium. Strikingly, this behavior has no optical signature, which agrees with experiments. 15,18,33 The computed optical signal precisely coincides with that computed in the absence of any coupling, i.e. for pores filling in the order of their diameters (Figure 9). This lack of optical signal follows from the fact that pores condense at equilibrium in the order of their diameter up to P3 , the lower closure pressure of the hysteresis loop. Because the sites are uncorrelated, the scattered intensity then corresponds to a random distribution of filled pores, and is very small. At T = 3 K, Figure 1 shows that P3 corresponds to a radius of 2.5 nm. For our distribution, more than 50% of the sites are then filled at P3 . This explains why the percolation of the filled sites at a lower fraction (≈ 31%) has no optical signature. Above P3 , the sites already filled at P3 act as germs for correlated condensation. Their density being very large, most larger pores condense at equilibrium, with no optical counterpart, due to the destructive interferences between selected clusters (see Supporting 24


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Information). The large difference between condensation and evaporation below Tco reflects that, for the chosen distribution, the germ density at the percolation point is large for condensation, and very small for evaporation. Here, this results from the fact that the smallest pores can condense at equilibrium, while the symmetrical process is forbidden by pore-blocking. This difference is however more general and follows from the fact that the spinodal instability for condensation has no counterpart for evaporation. Even for a distribution such that no isolated pore can fill at equilibrium, the smallest pores fill due to the spinodal instability and act as germs allowing some other pores to condense at equilibrium due to the coupling. At the percolation point (for condensation), the density of such germs is so large that no significant optical signal can be measured. Correlatively, due to this large density of germs, we show in the Supporting Information that a good approximation of the pore diameter distribution can be retrieved from the condensation branch by assuming the pores to fill at equilibrium (see Figure S11).

3.16 K (f) 3.16 K 3.40 K 3.75 K 4.34 K 4.60 K

0.2

V(R)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.1

0 0

2

4

6

8

10

12

pore radius R (nm)

Figure 10: Pore volume distribution inferred from a BJH-like analysis of the condensation branches of the experimental isotherms, assuming condensation to take place at equilibrium. Isotherm 3.16 K (f) is measured using the flowmeter described in the Supporting Information (§S2.2) instead of the temperature controlled reservoir.

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Determination of the pore volume distribution

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For Vycor, this major prediction of

our model is supported by our experimental observation that, on the condensation branch, the relaxation is much smaller than on the evaporation branch (see Supporting Information for a detailed discussion). We therefore determine the pore size distribution for Vycor from our condensation branches from 3.16 K up to Tch , using the BJH procedure to account for the contribution of the adsorbed film to the density change along the isotherm. The single pore properties (equilibrium pressure and average density at a given pressure) were obtained from our full single pore model. 26 Figure 10 shows the inferred distribution, again taking a van der Waals interaction strength of 1200 KÅ3 . Quite remarkably, it is identical over the full temperature range, which validates our assumption that condensation takes place at equilibrium. The distribution is asymmetrical, being flatter on the low radius side than on the large radius one. It peaks around 5 nm and extends up to radii of 8 nm. The resulting expected closure temperature, computed using our single pore model 26 is about 4.75 K when a barrier of 40 kB T can be overcome. This is close to the measured Tch ≈ 4.625 K. We therefore conclude that the pore size distribution derived from our multiscale approach is globally consistent. This conclusion might seems at odds with our above assumption of a mean pore diameter of 3 nm. However, we stress that we determine the pore volume distribution rather than the pore diameter distribution. For the assumed gaussian distribution, the pore volume distribution would peak between 4 nm and 4.2 nm if we assume that the average volume of a pore of radius R scales as R2 or R3 . This is closer to the observed 5 nm. Moreover, as discussed in the modelization part, our description of Vycor is oversimplified. We cannot thus expect the theoretical distribution accounting for the experimental temperature evolution to quantitatively agree with the actual one. Using the above determined experimental pore volume distribution, and assuming the pore volume to scale as the square of its radius, we can deduce the pore diameter distribution. The percolation threshold of 31% then corresponds to a value of Rperc between 3 and 4 nm, 26


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comparable to Rperc ≈3.7 nm for the assumed gaussian distribution. The predicted Tco is then around 3.35 K, while we observe the scattered intensity to increase between 3.16 K and 3 K. However, as above, this discrepancy can be explained by our simplified description of Vycor. A model with a lower coordinance would increase the 31% threshold. This would push the percolation pressure to lower values, hence decrease the predicted crossover temperature.

CONCLUSION In this paper, we presented an extensive experimental study of the condensation and evaporation of helium in a wide temperature range below the bulk critical temperature Tc . We also developed a lattice model including the effect of thermal activation at the single pore level and the collective effects resulting from the coupling between neighboring pores. This model allows to predict the isotherms and structure factor for any pressure history, including arbitrary scanning curves. Our main experimental findings are the demonstration of activated relaxation, which, in agreement with Machin’s theory, 41 explains the closure of the hysteresis loop below Tc , and the determination of the fluid spatial correlations during condensation and evaporation. These findings are consistently explained by the model, demonstrating that the evaporation mechanism smoothly evolves from pure invasion percolation from the sample’s surfaces at the lowest temperature to percolation from bulk cavitated sites at larger temperatures. Since a key ingredient of this crossover is a dependence of the cavitation pressure on the pore radius, the agreement of theory and experiment indirectly validates this prediction of our single pore description. Our collective model also reproduces the experimental lack of optical signal during condensation, a fact already recognized in several experiments, but never explained. Using percolation concepts (selection of Bernoulli clusters), we explain this lack of optical signature and show that it does not contradict our model’s prediction that condensation, as evaporation, is a collective process. An important feature of this prediction is that, under a wide range of conditions, condensation of pores

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should proceed at equilibrium. This leads us to suggest that, for disordered porous materials, the classical Barrett-Joyner-Halenda approach, when applied to the condensation branch using a properly extended version of the Kelvin equation, should give a good approximation of the pore size distribution in complex porous materials. This conclusion is supported by the application of this scheme to our experimental results, which yields the same pore volume distribution in a wide temperature range. To our knowledge, such an agreement had never been reported before. It would be highly valuable to further test this conclusion using other fluids (e.g. nitrogen) in Vycor, as well as other materials.

Acknowledgement We acknowledge ANR for funding the experimental work and M. Melich’s post-doctoral stay (ANR-06-BLAN-0098).

Supporting Information Available The free of charge Supporting Information pdf file gives details on our experimental and modeling methods. It also presents a larger set of relaxation data. Finally, it provides details on the analysis of the modelisation results.

References (1) Barrett, E. P.; Joyner, L. G.; Halenda, P. P. The Determination of Pore Volume and Area Distributions in Porous Substances .1. Computations from Nitrogen Isotherms. J. Am. Chem. Soc. 1951, 73, 373–380. (2) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkowiac, M. Phase Separation in Confined Systems. Rep. Prog. Phys. 1999, 62, 1573–1659.

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(3) Monson, P. Understanding Adsorption/Desorption Hysteresis for Fluids in Mesoporous Materials Using Simple Molecular Models and Classical Density Functional Theory. Microporous Mesoporous Mater. 2012, 160, 47 – 66. (4) Evans, R. Fluids Adsorbed in Narrow Pores: Phase Equilibria and Structure. J. Phys.: Condens. Matter 1990, 2, 8989–9007. (5) Neimark, A. V.; Ravikovitch, P. I.; Vishnyakov, A. Bridging Scales from Molecular Simulations to Classical Thermodynamics: Density Functional Theory of Capillary Condensation in Nanopores. J. Phys.: Condens. Matter 2003, 15, 347–365. (6) Saam, W. F.; Cole, M. W. Excitations and Thermodynamics for Liquid-Helium Films. Phys. Rev. B 1975, 11, 1086–1105. (7) Wallacher, D.; Kunzner, N.; Kovalev, D.; Knorr, N.; Knorr, K. Capillary Condensation in Linear Mesopores of Different Shape. Phys. Rev. Lett. 2004, 92, 195704. (8) Grosman, A.; Ortega, C. Cavitation in Metastable Fluids Confined to Linear Mesopores. Langmuir 2011, 27, 2364–2374. (9) Ravikovitch, P. I.; Neimark, A. V. Experimental Confirmation of Different Mechanisms of Evaporation from Ink-Bottle Type Pores: Equilibrium, Pore Blocking, and Cavitation. Langmuir 2002, 18, 9830–9837. (10) Morishige, K.; Tateishi, N. Adsorption Hysteresis in Ink-Bottle Pore. J. Chem. Phys. 2003, 119, 2301–2306. (11) Rasmussen, C. J.; Vishnyakov, A.; Thommes, M.; Smarsly, B. M.; Kleitz, F.; Neimark, A. V. Cavitation in Metastable Liquid Nitrogen Confined to Nanoscale Pores. Langmuir 2010, 26, 10147–10157.

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(12) Mason, G. A Model of Adsorption Desorption Hysteresis in which Hysteresis is Primarily Developed by the Interconnections in a Network of Pores. Proc. R. Soc. London, Ser. A 1983, 390, 47–72. (13) Mason, G. Determination of the Pore-Size Distributions and Pore-Space Interconnectivity of Vycor Porous Glass from Adsorption-Desorption Hysteresis Capillary Condensation Isotherms. Proc. R. Soc. London, Ser. A 1988, 415, 453–486. (14) Parlar, M.; Yortsos, Y. Percolation Theory of Vapor Adsorption-Desorption Processes in Porous Materials. J. Colloid Interface Sci. 1987, 124, 162–176. (15) Page, J. H.; Liu, J.; Abeles, B.; Herbolzheimer, E.; Deckman, H. W.; Weitz, D. A. Adsorption and Desorption of a Wetting Fluid in Vycor Studied by Acoustic and Optical Techniques. Phys. Rev. E 1995, 52, 2763–2777. (16) Parlar, M.; Yortsos, Y. Nucleation and Pore Geometry Effects in Capillary Desorption Processes in Porous Media. J. Colloid Interface Sci. 1989, 132, 425–443. (17) Morishige, K. Hysteresis Critical Point of Nitrogen in Porous Glass: Occurrence of Sample Spanning Transition in Capillary Condensation. Langmuir 2009, 25, 6221– 6226. (18) Bonnet, F.; Melich, M.; Puech, L.; Wolf, P. E. Light Scattering Study of Collective Effects during Evaporation and Condensation in a Disordered Porous Material. Europhys. Lett. 2013, 101, 16010. (19) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Sarkisov, L.; Tarjus, G. Capillary Condensation in Disordered Porous Materials: Hysteresis Versus Equilibrium Behavior. Phys. Rev. Lett. 2001, 87, 055701. (20) Detcheverry, F.; Kierlik, E.; Rosinberg, M.; Tarjus, G. Mechanisms for Gas Adsorption

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and Desorption in Silica Aerogels: The Effect of Temperature. Langmuir 2004, 20, 8006–8016. (21) Aubry, G. J.; Bonnet, F.; Melich, M.; Guyon, L.; Spathis, P.; Despetis, F.; Wolf, P.-E. Condensation of Helium in Aerogel and Athermal Dynamics of the Random-Field Ising Model. Phys. Rev. Lett. 2014, 113, 085301. (22) Woo, H. J.; Sarkisov, L.; Monson, P. A. Mean-Field Theory of Fluid Adsorption in a Porous Glass. Langmuir 2001, 17, 7472–7475. (23) Woo, H. J.; Porcheron, F.; Monson, P. A. Modeling Desorption of Fluids from Disordered Mesoporous Materials. Langmuir 2004, 20, 4743–4747. (24) Bonnet, F. Étude de la Condensation et de l’Évaporation de l’Hélium dans les Milieux Poreux : Effets du Confinement et du Désordre. Ph.D. thesis, Grenoble University, 2009. (25) Cross, B.; Puech, L.; Wolf, P. E. A Temperature-Controlled Device for Volumetric Measurements of Helium Adsorption in Porous Media. J. Low Temp. Phys. 2007, 148, 903. (26) Bonnet, F.; Wolf, P.-E. Thermally Activated Condensation and Evaporation in Cylindrical Pores. J. Phys. Chem. C 2019, 123, 1335–1347. (27) Guyer, R. A.; McCall, K. R. Capillary Condensation, Invasion Percolation, Hysteresis, and Discrete Memory. Phys. Rev. B 1996, 54, 18–21. (28) Puibasset, J. Monte-Carlo Multiscale Simulation Study of Argon Adsorption/Desorption Hysteresis in Mesoporous Heterogeneous Tubular Pores like MCM-41 or Oxidized Porous Silicon. Langmuir 2009, 25, 903–911. (29) Levitz, P.; Ehret, G.; Sinha, S.; Drake, J. Porous Vycor Glass - the Microstructure as

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Probed by Electron-Microscopy, Direct Energy-Transfer, Small-Angle Scattering, and Molecular Adsorption. J. Chem. Phys. 1991, 95, 6151–6161. (30) Hoshen, J.; Kopelman, R. Percolation and Cluster Distribution. I. Cluster Multiple Labeling Technique and Critical Concentration Algorithm. Phys. Rev. B 1976, 14, 3438–3445. (31) Here, we ignore the vectorial nature of the scattered field. This is justified in the Rayleigh regime if the incident polarization is perpendicular to the scattering plane. Both conditions are fulfilled in our experiments. (32) This procedure assumes the configuration to be statistically isotropic. While this is true for the simulation, this might not be so in an actual scattering experiment where the structure factor is measured locally for a given incident direction. In this case, comparison to the model requires to average over different configurations rather than over the full shell. (33) Soprunyuk, V.; Wallacher, D.; Huber, P.; Knorr, K.; Kityk, A. Freezing and Melting of Ar in Mesopores Studied by Optical Transmission. Phys. Rev. B 2003, 67 . (34) Thommes, M.; Smarsly, B.; Groenewolt, M.; Ravikovitch, P.; Neimark, A. Adsorption Hysteresis of Nitrogen and Argon in Pore Networks and Characterization of Novel Micro- and Mesoporous Silicas. Langmuir 2006, 22, 756–764. (35) Brown, A. J. The Thermodynamic and Hysteresis of Adsorption. Ph.D. thesis, School of Chemistry, University of Bristol, UK, 1963. (36) Quinn, H. W.; McIntosh, R. The Hysteresis Loop in Adsorption Isotherms on Porous Vycor Glass and Associated Dimensional Changes of the Adsorbent II. Can. J. Chem. 1957, 35, 745–756.

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(37) Brewer, D. F.; Champeney, D. C. Sorption of Helium and Nitrogen on Vycor Porous Glass. Proc. Phys. Soc., London 1962, 79, 855–868. (38) A difference in the pore size distributions between our two samples, which could also explain a different Tch , is not likely as the 77 K nitrogen sorption isotherm reported in Ref. 37 is quite similar to ours. (39) Burgess, C.; Everett, D.; Nuttall, S. Adsorption Hysteresis in Porous Materials. Pure Appl. Chem 1989, 61, 1845–1852. (40) Thommes, M.; Findenegg, G. Pore Condensation and Critical-Point Shift of a Fluid in Controlled-Pore Glass. Langmuir 1994, 10, 4270–4277. (41) Machin, W. D. Properties of Three Capillary Fluids in Critical Region. Langmuir 1999, 15, 169–173. (42) Ball, P.; R.Evans, Temperature Dependence of Gas Adsorption on a Mesoporous Solid: Capillary Criticality and Hysteresis. Langmuir 1989, 5, 714–723. (43) Valiullin, R.; Naumov, S.; Galvosas, P.; Kaerger, J.; Woo, H.-J.; Porcheron, F.; Monson, P. A. Exploration of Molecular Dynamics during Transient Sorption of Fluids in Mesoporous Materials. Nature 2006, 443, 965–968. (44) As shown by Figure 5a, the optical signal at θ = 45o and 3 K shows strong fluctuations. These fluctuations are uniform over the illuminated slice. They likely result from the fluctuations of the condensed mass due to the small oscillations of the room temperature induced by the air-conditioning system. (45) The larger fraction span for theory is explained by the fact that, for the empty pores, the present model neglects the adsorbed film. The more precise model developed in Ref. 26 for a single pore shows that this film significantly contributes to the liquid fraction.

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On the Mechanisms of Condensation and Evaporation in Disordered

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